Is "pinky swear" a childish expression? - No, but it is used mostly by children.
Do adults use this expression? - Yes, jokingly. Adults don't do pinky swears they just swear as in 'I swear I will do such and such.'
Context? Joking with friends, talking to children.
Examples? The only example I could think of goes something like this. It's between you and your close friend, you're a piano player and you're going to play publicly soon and you want her to come and cheer you up.
- (Your friend) I promise I will go to the concert no matter what.
- (You) Pinky promise?
You both laugh and maybe actually cross pinkies.